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/sci/ - Science & Math

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4599888 No.4599888 [Reply] [Original]

logicians, mathematicians, i've got a question

how can it be, that if a first-order theory has an infinite model of cardinality K, then it has infinite models of every cardinality lesser or greater than K (http://en.wikipedia.org/wiki/Löwenheim-Skolem_theorem)
..but at the same time, if two sets are of different infinite cardinalities, then there is no bijection between them??

for example, take peano's arithmetic and the theory of real numbers..
peano's arithmetic has an intended model of infinite cardinality aleph 0
the theory of real numbers has an intended model of infinite cardinality greater than aleph 0
however, given the lowenheim-skolem theorem, then peano's arithmetic also has a model of cardinality greater than aleph 0, and the theory of real numbers also has a model of cardinality aleph 0
however, there is no bijection between the naturals and the reals, thus, there is no bijection between the domain of the intended model of peano's arithmetic, and the domain of the intended model of the theory of reals

how can this be?? is this skolems paradox? how is it not a contradiction?

>> No.4599927

>Different
>Infinite
>No Bijection

How?

>> No.4599940
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4599940

>tfw no bijection

>> No.4599980
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4599980

nobody?

>> No.4600009
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4600009

last bump, don't let me down /sci/
any ideas? is the question clear?

>> No.4600038

skolem himself explained why its not a contradiction, i really dont know why we call it a paradox.

>> No.4600052
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4600052

>>4600038
i guess because it SEEMS like a contradiction..
but is my question the same as skolem's paradox? i read the statement of the paradox and it doesn't seem to be the same, but maybe they are and i just don't see it

>> No.4600072

>>4600052
im really sorry but its late and ill go to sleep now, hope you find someone who will actually read your question.
why not just ask some logic grad, theve got nothing to do with their time anyways?

>> No.4600080
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4600080

>>4600072
i just sent the question to one
good night

>> No.4600110
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4600110

another way of putting the question, which actually is the reason why i'm interested, is this:
how can it be that the reals are reducible to the naturals (that a model of the reals can be found in the naturals), if there is no way of making a correspondence between them?